Power Calculations for DNA Applications
Sample Size, Power and Effect Size
Below we provide estimates of power and effect size for various genetic tests and subgroups
within the Framingham Heart Study. Estimates are provided for the following situations:
1) linkage analyses with the 330 pedigrees included in the genome scan with microsatellite
markers, conducted by the Mammalian Genotyping Service,
2) association analyses for unrelated individuals with varying sample sizes
3) association analyses for family-based association tests
Linkage Analyses in 330 Pedigrees
All power calculations for linkage were performed using SOLAR v1.6.6 (Almasy and
Blangero, 1998). We evaluated the power of a likelihood-ratio test for linkage for a range of
QTL heritabilities (5% to 40%) and various sample sizes. The different sample sizes will reflect
the variation in the availability of data for different phenotypes. Three sample sizes, ranging
from the largest possible size (e.g. for a trait measured in all Framingham participants such as
blood pressure) to smaller sizes (e.g. traits measured only in the Offspring Cohort or a subset of
Offspring) were considered within the existing 330 pedigrees. For these samples we computed
power for three critical values (LOD scores 1.0, 2.0 and 3.0). These critical values encompass
LOD scores that are suggestive of linkage (LOD  1.0 or 1.5) or establish significant linkage
(LOD  3). Given a selected critical value, an approximate value of the power of the likelihoodratio test is computed using the non-central 2 distribution whose non-centrality parameter is
twice the difference between the expected log-likelihoods of the null and alternative hypotheses
of linkage (Sham et al., 2000). In this analysis the non-centrality parameters were estimated by
means of simulation using our pedigrees.
For each QTL heritability, 100 data sets were simulated using the 330 pedigree structure
and a genetic model containing a marker with 10 alleles of equal frequency (0.90 heterozygosity)
that was linked to a diallelic QTL with 10% allelic frequency. We evaluated other allele
frequencies, but found that all were essentially equivalent since the primary determinant of
power here is the proportion of variance that the QTL explains. Simulation was performed using
the “simqtl” command in SOLAR. First, the marker and the phenotypic data (under the assumed
QTL variance) were generated and the IBD probabilities were computed for all observations in
the pedigrees. For a given sample the non-centrality parameter of the 2 distribution was
estimated from a twopoint linkage analysis of that sample.
Tables 1 presents the expected lod score (ELOD, the average twopoint LOD score from
100 simulated data sets) and power computed for the various samples and levels of QTL
heritability in the 330 families. In these tables the three columns for power represent different
critical values. For example, the estimated power for sample size N=2885 in 330 pedigrees and
20% QTL heritability was 99%, 92% and 76% if linkage is deemed significant for LOD  1.0,
LOD  2.0 and LOD  3.0, respectively.
Table 1: Power for Linkage Analyses in 330 Pedigrees
Original Cohort and Offspring
Offspring
Offspring Subset
[N=2885, Ped=330]
[N=1672, Ped=330]
[N=1228, Ped=326]
QTL
Heritabilit ELOD LOD1 LOD2 LOD3 ELOD LOD1 LOD2 LOD3 ELOD LOD1 LOD2 LOD3
y
(%)
5
0.39
21
4
1
0.30
16
3
1
0.21
12
2
0
10
1.20
58
25
9
0.83
43
14
4
0.51
27
7
1
15
2.47
89
63
36
1.68
74
40
17
0.98
49
18
6
20
4.22
99
92
76
2.84
93
72
46
1.63
72
38
16
25
6.50
100
99
96
4.36
99
93
78
2.47
89
63
37
30
9.35
100
100
100
6.26
100
99
95
3.53
97
84
62
35 12.87
100
100
100
8.60
100
100
99
4.84
99
95
84
*For 40% QTL heritability, power greater than 95%
Association Studies with Biologically Unrelated Subjects
Table 2 displays detectable effect sizes (ΔR2 or percent variation explained by a QTL SNP) for a
quantitative trait measured on unrelated subjects for power of 80% or 90% and alpha levels of
0.01 0.001 and 0.0001. These calculations are performed for varying samples sizes, from 1800
to 800. The numbers for 1800 correspond to genotyping the unrelated plate set and all subjects
having the trait. Thus, for 1% significance level, if background (baseline) covariates explain
30% of the variation in the trait (R2 base), then a sample of size 1800 will have 80% power to
detect a QTL variance of 0.0054 and 90% power to detect a QTL variance of 0.0068.
Table 2: QTL variance sufficient for power 0.80 (0.90) in association analyses of unrelated subjects
Analysis with General Linear Model having unrestricted genotype means [u(AA), u(Aa), u(aa)]
Entries are values of QTL variance for stated power and significance level
given fixed sample size, and non-QTL baseline R-squared, with total variance=1.0
Sample
Size
Power
R2 base
1800
1800
1800
1800
1800
1800
1700
1700
1700
1700
1700
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
Significance Level
0.01
0.001
0.0001
0.0039
0.0054
0.0069
0.0048
0.0068
0.0087
0.0041
0.0057
0.0074
0.0051
0.0072
0.0055
0.0076
0.0098
0.0066
0.0092
0.0119
0.0058
0.0081
0.0104
0.0070
0.0098
0.0070
0.0098
0.0126
0.0083
0.0116
0.0149
0.0074
0.0104
0.0133
0.0087
0.0122
1700
1600
1600
1600
1600
1600
1600
1500
1500
1500
1500
1500
1500
1400
1400
1400
1400
1400
1400
1300
1300
1300
1300
1300
1300
1200
1200
1200
1200
1200
1200
1100
1100
1100
1100
1100
1100
1000
1000
1000
1000
1000
1000
900
900
900
900
900
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.8
0.8
0.8
0.9
0.9
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.5
0.3
0.0092
0.0043
0.0061
0.0078
0.0054
0.0076
0.0098
0.0046
0.0065
0.0083
0.0058
0.0081
0.0104
0.0050
0.0069
0.0089
0.0062
0.0087
0.0112
0.0053
0.0075
0.0096
0.0067
0.0094
0.0120
0.0058
0.0081
0.0104
0.0072
0.0101
0.0130
0.0063
0.0088
0.0114
0.0079
0.0111
0.0142
0.0069
0.0097
0.0125
0.0087
0.0122
0.0156
0.0077
0.0108
0.0139
0.0097
0.0135
0.0126
0.0061
0.0086
0.0110
0.0074
0.0104
0.0133
0.0065
0.0092
0.0118
0.0079
0.0111
0.0142
0.0070
0.0098
0.0126
0.0085
0.0119
0.0152
0.0075
0.0106
0.0136
0.0091
0.0128
0.0164
0.0082
0.0114
0.0147
0.0099
0.0138
0.0178
0.0089
0.0125
0.0160
0.0108
0.0151
0.0194
0.0098
0.0137
0.0177
0.0118
0.0166
0.0213
0.0109
0.0152
0.0196
0.0131
0.0184
0.0157
0.0079
0.0110
0.0142
0.0093
0.0130
0.0167
0.0084
0.0117
0.0151
0.0099
0.0139
0.0178
0.0090
0.0126
0.0162
0.0106
0.0148
0.0191
0.0097
0.0135
0.0174
0.0114
0.0160
0.0205
0.0105
0.0147
0.0188
0.0124
0.0173
0.0222
0.0114
0.0160
0.0206
0.0135
0.0189
0.0242
0.0126
0.0176
0.0226
0.0148
0.0207
0.0266
0.0139
0.0195
0.0251
0.0164
0.0230
900
800
800
800
800
800
800
0.9
0.8
0.8
0.8
0.9
0.9
0.9
0.1
0.5
0.3
0.1
0.5
0.3
0.1
0.0174
0.0087
0.0122
0.0156
0.0109
0.0152
0.0195
0.0236
0.0122
0.0171
0.0220
0.0148
0.0207
0.0266
0.0296
0.0157
0.0219
0.0282
0.0185
0.0258
0.0332
Differences between SNP genotypes relate to ΔR2 through allele frequencies (let f denote minor
allele frequency) ─ which determine expected genotype frequencies (g1, g2 and g3) ─ and
through covariate-adjusted genotype-specific means (μ1, μ2 and μ3). Without loss of generality,
we assume trait values are standardized to variance 1. Then ΔR2 = Σ gi(μi – μ)2 where μ= Σ giμi
is the overall mean. With an additive genetic model, μ2-μ1= θ and μ3-μ1= 2θ. Figure 1 displays
the relation between the increment ΔR2, the minor allele frequency, f, and the additive difference
between means, θ. For example, suppose we determine that we have power 0.80 to detect a SNP
that contributes 2% of the variance (Δ=0.02). If the minor allele frequency is f=0.10 then
θ0.33; note that θ is in standard deviation units. If the minor allele frequency is f=0.30, then
θ0.22. In conjunction with the power table, this Figure helps us to determine which
combinations of allele frequencies and genotype mean differences will be compatible with the
value of ΔR2 from power calculations for specified power and significance level.
Figure 1. Delta R-square related to Allele Frequency
Theta = difference between genotype means
0.5
Theta
Delta R-sq
0.005
0.01
0.015
0.02
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
Minor allele frequency
0.4
0.5
Association Studies using Family-Based Association Test
Table 3 displays power calculated using PBAT for the family-based association test with the
family plate set. Two sample sizes were evaluated: n=1398 from 400 nuclear families using
essentially all subjects on the plate set and n=1061 from 291 nuclear families, a subset of the
total. Power was computed for a range of QTL and marker minor allele frequencies. The QTL
and the marker are assumed to be in linkage disequilibrium (D’~1). Thus, when the minor allele
for the QTL is equal to that of the marker, it is assumed that the marker and the QTL are the
same and power is at its maximum. Although the D’ remains constant, as the differences
between the minor alleles for the QTL and the marker increase, the LD correlation (R2) and the
study power decrease.
For example, if a study used 1400 subjects on the family plate set for a trait with allele frequency
for the QTL of 5% and allele frequency of the nearby marker of 10%, the power to detect a QTL
variance (heritability) of 10% would be 52% for an alpha of 0.001 and 93% for an alpha of 0.05.
When the allele frequencies of the marker and the underlying functional variant of the QTL are
the same, power in these examples increase to a maximum of 98% and 100%.
Sample
Size
Table 3: Power for FBAT analyses
Allele
Allele
Freq for Freq for
Functional Typed
Variant
Marker
QTL H2 Alpha
Power
1400
5
5
5
5
10
10
0.001
0.05
0.001
0.05
66
98
98
100
1400
5
10
5
5
10
10
0.001
0.05
0.001
0.05
21
72
52
93
1400
5
20
5
5
10
10
0.001
0.05
0.001
0.05
6
41
16
65
1400
5
30
5
5
10
10
0.001
0.05
0.001
0.05
2
26
7
44
1400
10
5
5
5
0.001
0.05
24
77
10
10
0.001
0.05
65
97
1400
10
10
5
5
10
10
0.001
0.05
0.001
0.05
71
98
99
100
1400
10
20
5
5
10
10
0.001
0.05
0.001
0.05
22
72
57
94
1400
10
30
5
5
10
10
0.001
0.05
0.001
0.05
9
50
27
77
1400
20
5
5
5
10
10
0.001
0.05
0.001
0.05
6
43
21
73
1400
20
10
5
5
10
10
0.001
0.05
0.001
0.05
24
75
66
97
1400
20
20
5
5
10
10
0.001
0.05
0.001
0.05
74
98
99
100
1400
20
30
5
5
10
10
0.001
0.05
0.001
0.05
36
85
80
99
1400
30
5
5
5
10
10
0.001
0.05
0.001
0.05
3
28
9
51
1400
30
10
5
5
10
10
0.001
0.05
0.001
0.05
10
53
33
84
1400
30
20
5
5
10
10
0.001
0.05
0.001
0.05
39
87
86
99
1400
30
30
5
5
10
10
0.001
0.05
0.001
0.05
75
98
99
100
1061
5
5
5
5
10
10
0.001
0.05
0.001
0.05
29
85
72
99
1061
5
10
5
5
10
10
0.001
0.05
0.001
0.05
8
50
22
76
1061
5
20
5
5
10
10
0.001
0.05
0.001
0.05
2
27
7
45
1061
5
30
5
5
10
10
0.001
0.05
0.001
0.05
1
18
3
30
1061
10
5
5
5
10
10
0.001
0.05
0.001
0.05
9
54
28
84
1061
10
10
5
5
10
10
0.001
0.05
0.001
0.05
36
87
82
100
1061
10
20
5
5
10
10
0.001
0.05
0.001
0.05
9
51
27
78
1061
10
30
5
5
10
0.001
0.05
0.001
4
33
11
10
0.05
55
1061
20
5
5
5
10
10
0.001
0.05
0.001
0.05
2
28
7
50
1061
20
10
5
5
10
10
0.001
0.05
0.001
0.05
9
54
31
84
1061
20
20
5
5
10
10
0.001
0.05
0.001
0.05
40
88
86
100
1061
20
30
5
5
10
10
0.001
0.05
0.001
0.05
15
64
46
91
1061
30
5
5
5
10
10
0.001
0.05
0.001
0.05
1
19
3
33
1061
30
10
5
5
10
10
0.001
0.05
0.001
0.05
4
35
13
62
1061
30
20
5
5
10
10
0.001
0.05
0.001
0.05
17
66
53
93
1061
30
30
5
5
10
10
0.001
0.05
0.001
0.05
41
88
88
100